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Manifold - Motivational example: the circle
The circle is the simplest example of a topological manifold after Euclidean space itself. Consider, for instance, the circle of radius 1 with its centre at the origin. If x and y are the coordinates of a point on the circle, then we have x² + y² = 1.
Locally, the circle resembles a line, which is one-dimensional. In other words, only one coordinate is needed to describe the circle locally. Consider, for instance, the top part of the circle, for which the y-coordinate is positive (the yellow part in Figure 1). Any point in this part can be described by the x-coordinate. So, there is a continuous bijection χtop, which maps the yellow part of the circle to the open interval (−1, 1) by simply projecting onto the first coordinate:
Such a function is called a chart. Similarly, there are charts for the bottom (red), left (blue), and right (green) parts of the circle. Together, these parts cover the whole circle and the four charts form an atlas for the circle.
The top and right charts overlap: their intersection lies in the quarter of the circle where both the x- and the y-coordinates are positive. The two charts χtop and χright map this part bijectively to the interval (0, 1). Thus a function T from (0, 1) to itself can be constructed, which first inverts the yellow chart to reach the circle and then follows the green chart back to the interval:
Such a function is called a transition map.
The top, bottom, left, and right charts demonstrate that the circle is a manifold, but do not form the only possible atlas. Charts need not be geometric projections, and the number of charts is a matter of some choice. Consider the charts
and
Here s is the slope of the line through the variable point at coordinates (x,y) and the fixed pivot point (−1,0); t is the mirror image, with pivot point (+1,0). The inverse mapping from s to (x,y) is given by
it can easily be confirmed that x²+y² = 1 for all values of the slope s. These two charts provide a second atlas for the circle, with
Each chart omits a single point, either (−1,0) for s or (+1,0) for t, so neither chart alone is sufficient to cover the whole circle. It can be shown that no single chart can ever cover the full circle; even simple examples require the flexibility of manifolds and multiple charts.
Manifolds need not be connected (all in "one piece"): a pair of separate circles is also a topological manifold. Manifolds need not be closed: a line segment without its ends is a manifold. Manifolds need not be finite: a parabola is a topological manifold. Other topological manifold examples include a hyperbola and the locus of points on the cubic curve y² - x³ + x = 0, which are neither connected, nor closed, nor finite.
However, examples such as two touching circles that share a point to form a figure-8 are excluded: a satisfactory chart to one-dimensional Euclidean space cannot be constructed around the shared point. (A different view is taken in algebraic geometry, where complex points on the quartic curve ((x − 1)² + y² − 1)((x + 1)² + y² − 1) = 0, whose real points alone form a pair of circles touching at the origin, are considered.)
Viewed using calculus, the circle transition function T is simply a function between open intervals, to give a meaning to the statement that T is differentiable. That is, T, and the other transition maps, are differentiable on (0, 1). Therefore, with this atlas the circle is a differentiable manifold.
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 Adapted from the Wikipedia article "Motivational example: the circle", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
